Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology)

Fabel, Paul

doi:10.4064/ba59-1-9

Cited by 31 publications

(51 citation statements)

References 12 publications

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“…For example, consider the Hawaiian earring (HE). Fabel [9] proved that the product q×q : Ω(HE)× Ω(HE) −→ π (HE). Moreover, Fabel [10] showed that for each n ≥ 1 there exists a compact, path connected, metric space X such that multiplication is discontinuous in π qtop n (X, x).…”

Section: Resultsmentioning

confidence: 99%

“…Pakdaman et al [24] showed that for a locally (k − 1)-connected space X, π qtop k (X, x) is discrete if and only if X is semilocally n-connected at x (see also [13]). Fabel [9,10] and Brazas [3] presented some spaces for which their quasitopological homotopy groups are not topological groups. Moreover, despite of Fabel's result [9] that says the quasitopological fundamental group of the Hawaiian earring is not a topological group, Ghane et al [14] proved that the topological kth homotopy group of an k-Hawaiian like space is a prodiscrete metrizable topological group, for all k ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

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On topological shape homotopy groups

Nasri

Ghanei

Mashayekhy

et al. 2016

Topology and its Applications

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In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ), defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces which make them Hausdorff topological groups. We then exhibit an example in whichπ top k succeeds in distinguishing the shape type of X and Y whileπ k fails, for all k ∈ N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity ofπ top k with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the kth shape group induced by the kth shape loop space and show that it coincides with the above topology.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On topological shape homotopy groups

Nasri

Ghanei

Mashayekhy

et al. 2016

Topology and its Applications

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show abstract

“…It is known that π 1 (X, x 0 ) is a quasitopological group in the sense that inversion is continuous and the group operation is continuous in each variable. However, π 1 (X, x 0 ) can fail to be a topological group [4,13,14]. A general study of fundamental groups with the quotient topology appears in [7].…”

Section: Application To Topologized Fundamental Groupsmentioning

confidence: 99%

Realizing spaces as path-component spaces

Банах

Brazas

2020

Fund. Math.

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The path component space of a topological space X is the quotient space π 0 (X) whose points are the path components of X. We show that every Tychonoff space X is the path-component space of a Tychonoff space Y of weight w(Y) = w(X) such that the natural quotient map Y → π 0 (Y) = X is a perfect map. Hence, many topological properties of X transfer to Y. We apply this result to construct a compact space X ⊂ R 3 for which the fundamental group π 1 (X, x 0 ) is an uncountable, cosmic, k ω -topological group but for which the canonical homomorphism ψ : π 1 (X, x 0 ) → π 1 (X, x 0 ) to the first shape homotopy group is trivial.2010 Mathematics Subject Classification. Primary 55Q52, 58B05, 54B15; Secondary 22A05, 54C10, 54G15.

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“…There are several examples which illustrate that π qtop 1 (X, x 0 ) does not need to be a topological group even if X is a compact metric space (see [8,9]). Calcut and McCarthy [7] proved that the quotient topology induced by the compact-open topology of the fundamental group of a locally path connected and semilocally simply connected space is discrete which implies that π qtop 1 (X, x 0 ) is a topological group.…”

Section: On the Whisker Topology And Sltl Spacesmentioning

confidence: 99%

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

et al. 2018

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In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, π qtop 1 (X, x 0 ) and π wh 1 (X, x 0 ), respectively. In particular, we give necessary or sufficient conditions for coincidence and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the reverse of some of these implications do not hold, in general.

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Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology)

Cited by 31 publications

References 12 publications

On topological shape homotopy groups

On topological shape homotopy groups

Realizing spaces as path-component spaces

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

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